(a) From the defining property of the $\delta$ function,

$$\int_{-\infty}^{\infty} \delta(x) f(x) d x=f(0)$$

for any function $f$, prove that

(i) $\delta(-x)=\delta(x)$

(ii) $\delta(a x)=|a|^{-1} \delta(x)$ for $a \in \mathbb{R}, a \neq 0$,

(iii) If $g: \mathbb{R} \rightarrow \mathbb{R}, x \mapsto g(x)$ is smooth and has isolated zeros $x_{i}$ where the derivative $g^{\prime}\left(x_{i}\right) \neq 0$, then

$$\delta[g(x)]=\sum_{i} \frac{\delta\left(x-x_{i}\right)}{\left|g^{\prime}\left(x_{i}\right)\right|}$$

(b) Show that the function $\gamma(x)$ defined by

$$\gamma(x)=\lim _{s \rightarrow 0} \frac{e^{x / s}}{s\left(1+e^{x / s}\right)^{2}}$$

is the $\delta(x)$ function.